My first step on this rebuild was to clean all the grime from the engine. I found that the best way to do this is with a serious degreaser and a wire brush that resembles a toothbrush. You can get both these items from any auto parts store. Handling and working on a clean engine makes everything else go much smoother.
I had never worked on this type of engine before and it really became clear why things had been redesigned in newer Vespas. I only needed a couple of specialized tools (clutch puller, flywheel puller) to take the engine apart.
I originally intended to rebore the cylinder I got with the bike and fit a new piston but after closer inspection I found a small crack in the spigot. I realized that this barrel was useless unless I could get someone to weld it, but I found a second barrel for a great price at the local scooter shop instead.
You can see the hairline crack at the port cut out below.
I replaced the oil seals, gaskets, main bearings, rear wheel bearing, barrel, and slowly sourced out many of the missing parts over about 3 months. I even found a carb box (very rare) at First Kick Scooters who constantly amaze me with their parts selection. The carburetor was all in one piece and only needed very basic cleaning before re-assembly. If anyone needs them I have listed the standard jet sizes on the GS150 page.